motivation behind the notation $\ell^{\infty}$

Question

let $\ell^\infty$ be set of all bounded real sequences, why we use this notation $\ell^\infty$? inparticular $\ell$. when i suppose x to be an infinite sequence i feel need to write it an element of $R^{\infty}$ but i dont write so beacuse there must be some very particular reason for using $\ell^\infty$. One thing in my mind is boundedness but How boundedness is related to notation $\ell^\infty$

Answer 1

For any $1 \leq p < \infty$, we may define the $p$-norm of a sequence $x = (x_1, x_2, \ldots)$ of real or complex numbers, to be $$ \lVert x \rVert_p = (|x_1|^p + |x_2|^p + \cdots)^{\frac{1}{p}}$$ when the expression makes sense (i.e. the series on the right is convergent), and infinity otherwise. The space $\ell^p$ is then defined to be all sequences of numbers having finite $p$-norm. For example, an element $x$ of $\ell^1$ is precisely a sequence whose series $\sum_i x_i$ is absolutely convergent. An element of $\ell^2$ is what is called a "square-summable" sequence, and so on.

It is natural to wonder what $\ell^p$ is in the limit $p \to \infty$ (and if this makes sense). It turns out that in the limit, we have $$ \lim_{p \to \infty} \lVert x \rVert_p = \sup(|x_1|, |x_2|, \ldots) $$ where the expression on the right is called the "supremum norm" of $x$, which is finite if and only if the sequence $x$ is bounded. And so the space $\ell^\infty$ consists exactly of bounded sequences.

The Wikipedia page for $L_p$ space has a good overview of this. If you would like details on the proof of this limit, check this answer for them.